Aides to the Senate majority leader, Mitch McConnell of Kentucky, said it was unclear how long the delay would last.

The timing of the Senate vote is crucial. The more it is delayed, the more likely the bill is to fail, supporters and opponents say. Moreover, the Senate schedule will soon be packed with other legislation, like an increase in the statutory limit on federal borrowing and spending bills for the fiscal year that starts Oct. 1. In addition, Republicans are eager to cut taxes and simplify the tax code.

The Senate has struggled to pass a health care bill, delaying a vote on a previous version of the legislation in June.

Several Republican senators have expressed reservations or outright opposition to the new version as well, and Republicans need Mr. McCain’s vote to have any chance of passing it.

The bill, to repeal and replace major provisions of the Affordable Care Act, is a top priority for President Trump and Republicans in the House, which passed its own repeal bill in early May.

Mr. Cornyn acknowledged that “there’s uncertainty about what the final outcome will be.” Asked what would happen if the bill did not pass, he said: “I assume we’ll keep trying. But at some point, if Democrats won’t participate in the process, then we’re going to have to come up with a different plan.”

Critics of the Senate’s health care bill, taking advantage of the delay, said Sunday that Republican leaders needed to rework the legislation in fundamental ways. Given the additional time, they said, Senate committees should hold hearings to solicit opinions from the public and from experts on health care and insurance.

“We should not be making fundamental changes in a vital safety net program that’s been on the books for 50 years, the Medicaid program, without having a single hearing to evaluate what the consequences are going to be,” Senator Susan Collins, Republican of Maine, said on CNN’s “State of the Union.”

Roughly 20 million people have gained coverage through the Affordable Care Act, a pillar of President Barack Obama’s legacy. But Mr. Cornyn described the law on Sunday as a failed “exercise in central planning and command and control.”

The White House press secretary, Sean Spicer, declined to comment beyond wishing Mr. McCain a quick recovery, as did Marc Lotter, a spokesman for Vice President Mike Pence. Mr. Trump has been urging lawmakers to pass the bill, saying he is waiting with pen in hand.

On Friday, Mr. Pence assured skeptical governors that “the Senate health care bill strengthens and secures Medicaid for the neediest in our society,” putting the program, which serves more than 70 million low-income people, on “a path to long-term sustainability.”

But Ms. Collins said: “I would respectfully disagree with the vice president’s analysis. This bill would impose fundamental, sweeping changes in the Medicaid program, and those include very deep cuts. That would affect some of the most vulnerable people in our society, including disabled children, poor seniors. It would affect our rural hospitals and our nursing homes. And they would have a very difficult time even staying in existence.”

She added, “There are about eight to 10 Republican senators who have serious concerns about this bill.”

Republicans hold 52 Senate seats, and all Democratic senators oppose the bill. Ms. Collins and Senator Rand Paul, Republican of Kentucky, have said they will vote against even starting the debate, meaning all other Republican senators need to vote for the legislation if it is to pass.

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Mr. Paul’s reasons for opposing the bill are very different from Ms. Collins’s; he says it retains too much of the Affordable Care Act. And he predicted that support for the legislation would erode because of the delay prompted by Mr. McCain’s absence.

“The longer the bill’s out there, the more conservative Republicans are going to discover that it’s not repeal,” he said Sunday on CBS’s “Face the Nation.” “And the more that everybody’s going to discover that it keeps the fundamental flaw of Obamacare. It keeps the insurance mandates that cause the prices to rise, which chase young, healthy people out of the marketplace and leads to what people call adverse selection, where you have a sicker and sicker insurance pool and the premiums keep rising through the roof.”

Voters “elected us to repeal Obamacare,” Mr. Paul added. But with the bill drafted by Mr. McConnell, the senior senator from his home state, he said, “we’re going to keep most of the taxes, keep the regs, keep the subsidies and create a giant bailout superfund for the insurance companies.”

The Senate Democratic leader, Chuck Schumer of New York, said Sunday that he did not think delaying the vote would change the outcome.

“Time is not the problem in the present health care bill,” Mr. Schumer said. “The problem is the substance. It slashes Medicaid, which has become something that helps middle-class New Yorkers — millions of them, literally — and millions of Americans.”

The delay gives critics of the repeal bill more time to investigate numbers being used by the Trump administration to defend it.

The administration has been telling Congress and governors that the bill includes plenty of money to provide private insurance for people who would lose Medicaid coverage. But those estimates are based on particular assumptions chosen by administration officials. Gov. Brian Sandoval of Nevada, a Republican, and others have questioned the validity of those assumptions.

The assumptions, made by political appointees in the Trump administration, specify how states would use money provided by the bill and how many people losing Medicaid would buy private insurance.

In a report on the House bill last month, the office of the actuary at the federal Centers for Medicare and Medicaid Services said most of the people who lost Medicaid coverage would “ultimately be uninsured, though a small fraction would choose to purchase individual insurance.”

Millions of people have gained coverage in the 31 states that chose to expand Medicaid under the Affordable Care Act, and many of them are projected to lose that coverage under the Senate and House bills, which would roll back the expansion of Medicaid.

When Congress convened in January, Republicans appeared to be on course to repeal the Affordable Care Act within a month or two, but they met with growing resistance as lawmakers, consumers, doctors, hospitals and insurance companies scrutinized the proposals. Mr. McConnell delayed a vote scheduled for the week before the Fourth of July. Then, with no visible progress toward agreement, he delayed the Senate’s August recess by two weeks so senators could keep working.

Administration officials will use the time provided by the latest delay to try to persuade undecided Republican senators to vote for the bill. They will also try to raise doubts about the work of the Congressional Budget Office, which estimated that an earlier version of Mr. McConnell’s bill could increase the number of uninsured Americans by 22 million by 2026, compared with current law.

The nonpartisan budget office had been expected to issue a report on the latest draft of the bill on Monday, but it now plans to take more time.

Lawmakers are eager to see what the office says about a proposal added to the bill last week in a bid for support from the most conservative Republican senators. Under the proposal, insurers could offer cheaper, less comprehensive health plans if they also offered three standard plans with all the benefits required by the Affordable Care Act.

The author of the proposal, Senator Ted Cruz, Republican of Texas, said it would give consumers “the freedom to choose among more affordable plans” that were “free from Obamacare’s insurance regulations.”

The skimpier plans would cover less and presumably cost less, and insurers said they would also attract healthier people.

“These junk insurance plans could charge people more or simply deny them coverage based on pre-existing conditions,” said Senator Maria Cantwell, Democrat of Washington.